Exim send mail question

Is it possible to change the common name (mail.yourdomain.com) for sending email with exim in something different like for example: smtp or anything else?

This because we don't give out the send mail details but only the one to receive email (mail.yourdomain.com) as we had many relay problems with ISPs.
Recently someone found out by trying when he had problems with his own ISP that using for both receive and send email mail.yourdomain.com was doing the trick.